2. The force (not just its magnitude) due to a solid body being compressed/stretched in the direction is given by Fe = EA L AL where E is the object's Young's modulus, A is its cross-sectional area, L is its resting length, and AL is the resulting change in the the object's length due to the force Fe. From the similarities between Fe and Hooke's law, write down the expression for the potential energy stored in a stressed solid body.
2. The force (not just its magnitude) due to a solid body being compressed/stretched in the direction is given by Fe = EA L AL where E is the object's Young's modulus, A is its cross-sectional area, L is its resting length, and AL is the resulting change in the the object's length due to the force Fe. From the similarities between Fe and Hooke's law, write down the expression for the potential energy stored in a stressed solid body.
Engineering Fundamentals: An Introduction to Engineering (MindTap Course List)
5th Edition
ISBN:9781305084766
Author:Saeed Moaveni
Publisher:Saeed Moaveni
Chapter6: Fundamental Dimensions And Units
Section: Chapter Questions
Problem 13P
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![### Understanding the Force in a Compressed or Stretched Solid Body
**2.** The force (not just its magnitude) due to a solid body being compressed/stretched in the \( x \) direction is given by
\[ F_e = -\frac{EA}{L} \Delta L \]
where:
- **E**: the object's Young's modulus,
- **A**: its cross-sectional area,
- **L**: its resting length,
- **\(\Delta L\)**: the resulting change in the object's length due to the force \( F_e \).
From the similarities between \( F_e \) and Hooke’s law, we can derive the expression for the **potential energy** stored in a stressed solid body.
### Explanation:
1. **Young’s Modulus (E)**: This is a measure of the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a linear elastic material.
2. **Cross-sectional Area (A)**: The area of the section perpendicular to the force applied. In this context, it represents the area over which the force is distributed.
3. **Resting Length (L)**: The original length of the object before any force is applied.
4. **Change in Length (\(\Delta L\))**: The difference between the object’s original length and its length after the force is applied.
### Task
From the given equation \( F_e = -\frac{EA}{L} \Delta L \) and its similarities with Hooke’s Law, the goal is to write down the expression for the potential energy stored in a stressed solid body.
### Hooke's Law
Hooke's Law states that the force needed to extend or compress a spring by some distance \( x \) scales linearly with respect to that distance. Mathematically, it is given by:
\[ F = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.
### Potential Energy in Hookean Systems
The potential energy (U) stored in the system due to the deformation is given by:
\[ U = \frac{1}{2} k x^2 \]
### Potential Energy in a Stressed Solid Body
Rewriting the given force in the form of Hooke’s law, by analogy, we can determine the potential energy stored in the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F701927fd-ceaf-4fad-8338-7d1e3c22653c%2F049f5f16-4f93-452b-9bec-e90c1c2e4c95%2F7kcqfxb_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Force in a Compressed or Stretched Solid Body
**2.** The force (not just its magnitude) due to a solid body being compressed/stretched in the \( x \) direction is given by
\[ F_e = -\frac{EA}{L} \Delta L \]
where:
- **E**: the object's Young's modulus,
- **A**: its cross-sectional area,
- **L**: its resting length,
- **\(\Delta L\)**: the resulting change in the object's length due to the force \( F_e \).
From the similarities between \( F_e \) and Hooke’s law, we can derive the expression for the **potential energy** stored in a stressed solid body.
### Explanation:
1. **Young’s Modulus (E)**: This is a measure of the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a linear elastic material.
2. **Cross-sectional Area (A)**: The area of the section perpendicular to the force applied. In this context, it represents the area over which the force is distributed.
3. **Resting Length (L)**: The original length of the object before any force is applied.
4. **Change in Length (\(\Delta L\))**: The difference between the object’s original length and its length after the force is applied.
### Task
From the given equation \( F_e = -\frac{EA}{L} \Delta L \) and its similarities with Hooke’s Law, the goal is to write down the expression for the potential energy stored in a stressed solid body.
### Hooke's Law
Hooke's Law states that the force needed to extend or compress a spring by some distance \( x \) scales linearly with respect to that distance. Mathematically, it is given by:
\[ F = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.
### Potential Energy in Hookean Systems
The potential energy (U) stored in the system due to the deformation is given by:
\[ U = \frac{1}{2} k x^2 \]
### Potential Energy in a Stressed Solid Body
Rewriting the given force in the form of Hooke’s law, by analogy, we can determine the potential energy stored in the
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